96 Cosmological Models
The larger the ratio푀∕푟of the star, the higher is the required escape velocity.
Ultimately, in the ultra-relativistic case when푣=푐, only light can escape. At that
point a nonrelativistic treatment is no longer justified. Nevertheless, it just so hap-
pens that the equality in Equation (5.71) fixes the radius of the star correctly to be
precisely푟c, as defined above. Because nothing can escape the interior of푟c, not even
light,John A. Wheelercoined the term black hole for it in 1967. Note that the escape
velocity of objects on Earth is 11kms−^1 , on the Sun it is 2. 2 × 106 kmh−^1 ,butona
black hole it is푐.
This is the simplest kind of aSchwarzschild black hole,and푟cdefines its event
horizon. Inserting푟cinto the functions퐴and퐵,theSchwarzschild metricbecomes
d휏^2 =(
1 −
푟c
푟)
d푡^2 −(
1 −
푟c
푟)− (^1) d푟 2
푐^2
. (5.72)
Note that The Schwarzschild metric resembles the de Sitter metric [Equa-
tion (5.62)]. In the Schwarzschild metric the coefficient of d푡^2 is singular at푟=0,
whereas the coefficient of d푟^2 is singular at푟=푟c. However, if we make the transfor-
mation from the radial coordinate푟to a new coordinate푢defined by
푢^2 =푟−푟c,the Schwarzschild metric becomes
d휏푟=푢^2
푢^2 +푟cd푡^2 − 4 (푢^2 +푟c)d푢^2.The coefficient of d푡^2 is still singular at푢^2 =−푟c, which corresponds to푟=0, but
the coefficient of d푢^2 is now regular at푢^2 =0.
5.4 Black Holes
A particularly fascinating and important case is a black hole, a star of extremely high
density. Black holes are certainly the most spectacular prediction of general relativity,
and they appear to be ubiquitous in the nuclei of bright and active galaxies.
The Schwarzschild metric has very fascinating consequences. Consider a space-
craft approaching a black hole with apparent velocity푣=d푟∕d푡in the fixed frame of
an outside observer. Light signals from the spacecraft travel on the light cone, d휏=0,
so that
d푟
d푡=푐
(
1 −
푟c
푟)
. (5.73)
Thus the spacecraft appears to slow down with decreasing푟, finally coming to a
full stop as it reaches the mathematical singularity of d푡at theevent horizon푟=푟cin
the expression
푐d푡=
d푟
1 −푟c∕푟. (5.74)
Thetimeintervalsd푡between successive crests in the wave of the emitted light
become longer, reaching infinite wavelength at the singularity. Although the velocity